Re: Replacement of Cardinality (infinite middle)

On 8/19/2024 3:27 PM, Ross Finlayson wrote:

On 08/18/2024 09:56 PM, Jim Burns wrote:

[...]

>
I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering

...in both directions...

and that all the orderings of the set are well-orderings.

...in both directions...

That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.

...for comfortably more than a century.

Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?

They mostly agree.
Given the Axiom of Choice
(let us say, if an inaccessible cardinal exists),
they completely agree.
My impression from somewhere is that,
if they disagree,
they disagree on some very weird sets.
https://en.wikipedia.org/wiki/Finite_set

[...] is another little fact of mathematics
missing from your neat little hedgerow.

I mark my neat little hedgerow, and
I describe what's true everywhere inside the hedgerow.
That allows me to learn about
what's inside the hedgerow,
even though it's infinite and I am finite.
I like doing that.
It isn't wrong for me to do that.
I will continue doing that.